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Properties

Label 412698.cr
Number of curves $2$
Conductor $412698$
CM no
Rank $0$
Graph

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Show commands: SageMath
Copy content sage:E = EllipticCurve("cr1") E.isogeny_class()
 

Rank

Copy content sage:E.rank()
 

The elliptic curves in class 412698.cr have rank \(0\).

Complex multiplication

The elliptic curves in class 412698.cr do not have complex multiplication.

Modular form 412698.2.a.cr

Copy content sage:E.q_eigenform(10)
 
\(q + q^{2} + q^{3} + q^{4} + q^{6} + 2 q^{7} + q^{8} + q^{9} + q^{11} + q^{12} + 2 q^{14} + q^{16} + 2 q^{17} + q^{18} + 8 q^{19} + O(q^{20})\) Copy content Toggle raw display

Isogeny matrix

Copy content sage:E.isogeny_class().matrix()
 

The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the LMFDB numbering.

\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)

Isogeny graph

Copy content sage:E.isogeny_graph().plot(edge_labels=True)
 

The vertices are labelled with LMFDB labels.

Elliptic curves in class 412698.cr

Copy content sage:E.isogeny_class().curves
 
LMFDB label Cremona label Weierstrass coefficients j-invariant Discriminant Torsion structure Modular degree Faltings height Optimality
412698.cr1 412698cr1 \([1, 0, 0, -6243793, 5783062505]\) \(5577108481460841625/233729407061568\) \(1128167205569439976512\) \([2]\) \(27371520\) \(2.8044\) \(\Gamma_0(N)\)-optimal
412698.cr2 412698cr2 \([1, 0, 0, 3010647, 21461934753]\) \(625234740274982375/41585929145369928\) \(-200727337072233876799752\) \([2]\) \(54743040\) \(3.1509\)